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On *-representations of the Hopf *-algebra associated with the quantum group $$U_ q(n)$$. (English) Zbl 0721.17014
The paper under review is concerned with the quantum group $$U_ q(n)$$, i.e. the “compact form” of $$\text{GL}_ q(n)$$, from the viewpoint of the “algebra of continuous functions” on it. The author defines the algebra of rational functions on $$\text{GL}_ q(n)$$, and calls $$\text{Pol}(U_ q(n))$$ this algebra provided with a star operation analogous to the Cartan involution. He proves a sort of Poincaré-Birkhoff-Witt theorem for $$\text{Pol}(U_ q(n))$$, whose proof indeed does not involve the star operation. This enables him to define a triangular decomposition on $$\text{Pol}(U_ q(n))$$ (in fact, one triangular decomposition for each element $$\mu$$ of the symmetric group $${\mathfrak S}_ n)$$; in turn, this makes it possible to introduce Verma modules $$V^{\mu}(v)$$, for any $$\mu$$ in $${\mathfrak S}_ n$$, $$v \in {\mathbb C}^ n$$. Each $$V^{\mu}(v)$$ has a unique irreducible quotient, denoted by $$L^{\mu}(v)$$. One of the main results in the paper states that the list $L^{\mu}(v),\quad v=(v_ 1,\dots,v_ n),\quad | v_ i| =q^{I(\mu,i)},\quad \mu \in\mathfrak S_ n$ yields all the mutually inequivalent irreducible representations of $$\text{Pol}(U_ q(n))$$. (Here $$I(\mu, i)$$ is a well-defined integer.) (Similar results were obtained by Ya. S. Soibel’man [Dokl. Akad. Nauk SSSR 307, No. 1, 41–45 (1989; Zbl 0698.22015)], as remarked at the end of the article.)
Next, the author considers the $$C^*$$-completion of the $${}^*$$-algebra $$\text{Pol}(U_ q(n))$$, denoted $${\mathfrak C}_ n(U_ q(n))$$. (The classification stated above permits him to prove that the seminorm used in the completion is in fact a norm.) He proves that $${\mathfrak C}(U_ q(n))$$ is a “compact matrix pseudogroup” in the sense of Woronowicz and a type I $$C^*$$-algebra. For related material see [M. Rosso, Astérisque, Séminaire Bourbaki, Vol. 1990/91, Exp. No. 744, Astérisque 201-203, 443–483 (1991; Zbl 0752.17016)] and references therein; the reader interested in the representation theory of compact matrix pseudogroups can consult S. Z. Levendorskii and Ya. S. Soibel’man [Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Commun. Math. Phys. 139, No. 1, 141–170 (1991; Zbl 0729.17011)].

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 46L85 Noncommutative topology 46L87 Noncommutative differential geometry
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